5.7 Arithmetic and Geometric Sequences 277 We can use the following formula to determine the sum of the first n terms in an arithmetic sequence. Sum of the First n Terms in an Arithmetic Sequence The sum of the first n terms of an arithmetic sequence can be determined with the following formula where a1 represents the first term and an represents the nth term. s n a a ( ) 2 n n 1 = + In this formula, sn represents the sum of the first n terms, a1 is the first term, an is the nth term, and n is the number of terms in the sequence from a1 to a .n Example 5 Determining the Sum of an Arithmetic Sequence Determine the sum of the first 25 even natural numbers. Solution The sequence we are discussing is … 2, 4, 6, 8, 10, , 50 In this arithmetic sequence the first term is 2, the 25th term is 50, and there are 25 terms; therefore, = = a a 2, 50, 1 25 and n 25. = Thus, the sum of the first 25 terms is = + = + = = = s n a a s ( ) 2 25(2 50) 2 25(52) 2 1300 2 650 n n 1 25 The sum of the first 25 even natural numbers, + + + + + 2 4 6 8 50, is 650. 7 Now try Exercise 25 RECREATIONAL MATH The Martingale System The Martingale system is a strategy used by gamblers to determine the amount to bet while playing multiple rounds of a game of chance. The strategy requires the player to consistently bet a standard amount following a win and to double the previous bet following a loss. The underlying principle is that the first win will recover all the previous losses, thus ensuring that the player does not lose money in the long run. Let’s use a standard bet of $1 as an example. In the first round, the player bets $1. If the player wins, the bet in the second round remains $1. If the player loses, the bet in the second round doubles to $2. Next, if the player wins in the second round, the bet in the third round returns to $1; however, if the player loses in the second round, the bet in the third round doubles to $4. The system continues with the player betting $1 following a win, and doubling the previous bet following a loss. Although the strategy may work in the short term, a player may have difficulty doubling the bet after several consecutive losses. Casinos generally place a maximum on the amount a person can bet. Exercise 74 demonstrates why the Martingale system is a risky wagering strategy. Geometric Sequences The next type of sequence we will discuss is the geometric sequence. A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant. This constant is called the common ratio . The common ratio, r, can be determined by taking any term except the first and dividing that term by the preceding term. Examples of geometric sequences Common ratios … 2, 4, 8, 16, 32, r r 4 2 2; 8 4 2 = ÷ = = ÷ = − − − … 3, 6, 12, 24, 48, r r 6 ( 3) 2; ( 12) 6 2 = ÷ − = − = − ÷ = − … 2 3 , 2 9 , 2 27 , 2 81 , r r 2 9 2 3 2 9 3 2 1 3 ; 2 27 2 9 2 27 9 2 1 3 = ÷ = ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ = = ÷ = ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ = To construct a geometric sequence when the first term, a ,1 and common ratio, r, are known, multiply the first term by the common ratio to get the second term. Then multiply the second term by the common ratio to get the third term, and so on. Serpeblu/Shutterstock
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